652 BELL SYSTEM TECHNICAL JOURNAL 



of (34) which gives: 



{Wa - W,), =^K''T' +\KaaT' -{-^KuaT^- +^aa'T- -^ aaU^T. (58) 



Putting in the initial conditions we have 



(W. - Wa)o=Ik'T,' (59) 



and 



(Wb - W^)o ^lK-T,'[4gh{l + ¥) + igVi-' + 2}f- + h'l (60) 



From these, and placing the cathode potential equal to zero, we get 



4 



^ = 2g/^ + 1 + hK 



With (57) the ratio g may be eliminated from this, giving 



4 



k-M-^+h'- (*» 



When h is small, as it normally is, the last term may be disregarded, 

 giving 



h = , ' (approximately). (61a) 



i + 4VJVA 



Equations (57) and (61) allow g and h to be found in terms of Vp, 

 which can be measured directly, and Va which can be found in terms 

 of the current and the distance x^ by combining (54) and (59) to give: 



(Wc - WA)o = lK'"{6x,yi\ (62) 



This is the well-known Child's equation, and when W and K are 

 expressed in terms of voltage and current appears in the usual form 



Jo = - 2.34 X 10-^Va"Vxc' amperesjcm.^ (63)* 



Several other expressions which are useful for computation purposes 

 may be found from these d.-c. relations: 



* The negative sij^n occurs because of the assumed current direction, from the 

 cathode. The numerical factor is 2.34 instead of 2.33 given by Compton and 

 Langmuir " because of the value of e/m which was used, q.v. 



